Course Code:
MATH 439
Semester:
Fall
Course Type:
Core
P:
3
Lab:
0
Laboratuvar Saati:
0
Credits:
3
ECTS:
6
Course Language:
English
Course Objectives:
To develop the necessary background for modern analysis courses to follow
Course Content:

Basic concepts about topological spaces and metric spaces. Complete metric spaces, Baire’s theorem, Contracting mapping theorem and its applications. Compact spaces,  Arzela-Ascoli  Theorem Seperability, second countability, Urysohn's lemma and the Tietze extension theorem, Connected spaces, Weierstrass approximation theorem

Course Methodology:
1: Lecture, 2: Problem Solving
Course Evaluation Methods:
A: Written examination, B: Homework

## Vertical Tabs

### Course Learning Outcomes

 Learning Outcomes Program Learning Outcomes Teaching Methods Assessment Methods 1) Learns basic concepts of topological spaces with emphasis on metric spaces 1,2 A 2) Learns Cauchy sequences and completeness 2,2 A 3) Learns the concept of compact space 1,2 A 4) Learns Baier’s category 1,2 A 5) Learns Ascoli-Arzela theorem,  Weierstrass approximation 1,2 A 6) Acquires the skill of applying these concepts 1,2 A

### Course Flow

 Week Topics Study Materials 1 Basic concepts about metric spaces and examples 2 Open, closed sets, topology and convergence 3 Cauchy sequences and complete metric spaces,  Baire's theorem 4 Continuity and uniformly continuity,  spaces of continuous functions, Euclidean space 5 Contracting mapping theorem and its applications 6 The definition of topological spaces and some examples , elementary concepts, Open bases and open subbases 7 Compact spaces, Products of spaces,Tychonoff's theorem and locally compact spaces 8 Compactness for metric spaces 9 Arzela-Ascoli  Theorem 10 Seperability, second countability 11 Hausdorff spaces, Completely regular spaces and normal spaces 12 Urysohn's lemma and the Tietze extension theorem 13 Connected spaces, The components of a space, Totally disconnected spaces, Locally connected spaces 14 The Weierstrass approximation theorem ,The Stone-Weierstrass theorems

### Recommended Sources

 Textbook 1. S. Kumaresan, Topology of Metric Spaces 2. George F. Simmons, Topology and Modern Analysis 3. W A Sutherland, Introduction to Metric and Topological Spaces 4. E T Copson, Metric Spaces Additional Resources

### Material Sharing

 Documents Assignments Exams

### Assessment

 IN-TERM STUDIES NUMBER PERCENTAGE Mid-terms 2 100 Quizzes - 0 Assignments - 0 Total 100 CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE 60 CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE 40 Total 100

 COURSE CATEGORY Expertise/Field Courses

### Course’s Contribution to Program

 No Program Learning Outcomes Contribution 1 2 3 4 5 1 The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamenral research fields in mathematics (i.e., analysis, algebra, differential equations and geometry) X 2 Acquiring fundamental knowledge on fundamental research fields in mathematics X 3 Ability form and interpret the relations between research topics in mathematics X 4 Ability to define, formulate and solve mathematical problems X 5 Consciousness of professional ethics and responsibilty X 6 Ability to communicate actively X 7 Ability of self-development in fields of interest X 8 Ability to learn, choose and use necessary information technologies X 9 Lifelong education X

### ECTS

 Activities Quantity Duration (Hour) Total Workload (Hour) Course Duration (14x Total course hours) 14 3 42 Hours for off-the-classroom study (Pre-study, practice) 14 4 56 Mid-terms (Including self study) 2 15 30 Quizzes - Assignments - Final examination (Including self study) 1 20 20 Total Work Load 148 Total Work Load / 25 (h) 5.92 ECTS Credit of the Course 6